Erd\H{o}s-Moser Equation in Arithmetic Progressions

We consider the Erd\H{o}s-Moser equation $1^k+2^k+\cdots +(m-1{)}^k=m^k$ in arithmetic progressions. We prove among other things that when $k=2$, for any solution to exist, the above sum in arithmetic progression must consist of two or four terms. In either case, there are infinitely many solutions that can be completely characterized.